Question

Statement-1: If in a triangle orthocentre, circumcentre and centroid are rational points then its vertices must also be rational points. because
Statement-2: If the vertices of a triangle are rational point then the centroid, circumcentre and orthocentre are also rational points.

• A. Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
• B. Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
• C. Statement-1 is true, statement-2 is false.
• D. Statement-1 is false, statement-2 is true.

Answer 1

Answer: (D)

$S-2$ is obviously correct.
For statement-1: Let the circumcentre be at $(0,0)$ and the vertices of the triangle be $\left(x_1,y_1\right),\left(x_2,y_2\right)$ and $\left(x_3,y_3\right)$ then centroid is $\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}$ and orthocenre of the triangles becomes $\left(x_1+x_2+x_3,y_1+y_2+y_3\right)$.
This implies that if the centroid is rational then orthocentre is also rational but $\left(x_1+x_2+x_3\right)$ can be rational even if $x_1,x_2,x_3$ are not all rational.
e.g. $A(1,0);B(-1/2,\sqrt{3}/2);C(-1/2,-\sqrt{3}/2)$ where $G/H/C\cdot C$ are at $(0,0)$ i.e. rational points